E l e c t ro n ic
Jo u r n
a l o
f P
r o b
a b i l i t y Vol. 9 (2004), Paper no. 17, pages 544-559.
Journal URL
http://www.math.washington.edu/˜ejpecp/
Asymptotic Distributions and Berry-Esseen Bounds for Sums of
Record Values
ByQi-Man Shao 1, Chun Su 2 and Gang Wei
University of Oregon and National University of Singapore, University of Science and Technology of China, Hong Kong Baptist University
Abstract. Let{Un, n≥1}be independent uniformly distributed random variables, and{Yn, n≥
1}be independent and identically distributed non-negative random variables with finite third moments. Denote Sn = Pni=1Yi and assume that (U1,· · ·, Un) and Sn+1 are independent for every fixed n. In
this paper we obtain Berry-Esseen bounds forPn
i=1ψ(UiSn+1), whereψis a non-negative function. As
an application, we give Berry-Esseen bounds and asymptotic distributions for sums of record values.
Key Words and Phrases: Central limit theorem, Berry-Esseen bounds, sum of records, insurance risk
AMS 2000 Subject Classification: Primary 60F10, 60F15;
Submitted to EJP on December 1, 2003 Final version accepted on June 9, 2004.
1Research is partially supported by the National Science Foundation under Grant No. DMS-0103487 and grants
R-146-000-038-101 and R-1555-000-035-112 at the National University of Singapore
1
Introduction
Our interest in asymptotic distributions in this paper grew out of problems on sums of record values in insurance. Assume that the claim sizes{Xi, i≥1}are independent and identically distributed (i.i.d.)
positive random variables with a common distribution function F. Xk is called a record value if Xk> max
1≤i≤k−1Xi.
By convention, X1 is a record value and denoted by X(1). Define
L1 = 1, Ln= min{k > Ln−1, Xk > XLn−1}forn≥2.
(1.1)
{Ln, n≥1}is called the “record occurrence time” sequence of {Xn, n≥1}. Let X(n)=XLn.
(1.2)
Then {X(n), n ≥ 1} is the record sequence of {Xn, n ≥ 1}. According to insurance theory, what
leads to bankruptcy of an insurance company is usually those large claims that come suddenly. Thus, studying the laws of large claims is of significant importance for insurance industry. The limiting properties of the record values have been extensively studied in literature. Tata (1969) obtained a necessary and sufficient condition for the existence of non-degenerate limiting distribution for the standardizedX(n). de Haan and Resnick (1973) found almost sure limit points for record valuesX(n).
Arnold and Villasenor (1998) recently studied the limit laws for sums of record values
Tn= n
X
i=1 X(i).
(1.3)
When Xi has an exponential distribution, Arnold and Villasenor proved that a standardized Tn is
asymptotic normal. We refer to Embrechts, Kl¨uppelberg and Mikosch (1997), Mikosch and Nagaev (1998), Su and Hu (2002) for recent developments in this area.
A key observation in those proofs is that (see Tata (1969) and Resnick (1973)) if F is an expo-nential distribution with mean one, then {X(n), n ≥1} and {G
n := Pni=1Ei, n ≥ 1} have the same
distribution, where Ei are independent exponentially distributed random variables with mean one.
For a general random variable X with a continuous distribution function F, it is well-known that
F(X) is uniformly distributed on (0,1) and R(X) :=−ln(1−F(X)) has an exponential distribution with mean 1. Write
F−1(x) = inf{y: F(y)≥x}, ψ(x) = inf{y: R(y)≥x}forx≥0.
Then ψ(x) =F−1(1−e−x), and {X(n), n≥1} and {ψ(R(X(n))), n≥1} have the same distribution. Thus, we have
{X(n), n≥1}=d. {ψ(Gn), n≥1}
and in particular,
Tn= n
X
i=1
X(i) =d.
n
X
i=1
ψ(Gi) = n
X
i=1
ψ³ Gi Gn+1
Gn+1
´
Let {Ui, i≥ 1} be i.i.d uniformly distributed random variables on (0,1) independent of {Ei, i≥ 1},
and letUn,1 ≤Un,2 ≤ · · · ≤Un,n be the order statistics of {Ui,1≤i≤n}. It is known that
{Un,i,1≤i≤n}=d. {Gi/Gn+1,1≤i≤n}
and that{Gi/Gn+1,1≤i≤n}and Gn+1 are independent. Therefore Tn=d.
n
X
i=1
ψ(Un,iGn+1) = n
X
i=1
ψ(UiGn+1).
(1.4)
The main purpose of this paper is to study the asymptotic normality and Berry-Esseen bounds for the standardized Tn and solve an open problem posed by Arnold and Villasenor (1998).
This paper is organized as follows. The main results are stated in the next section, an application to the sum of record values is discussed in Section 3, and proofs of main results are given in Section 4. Throughout this paper, f(x) ∼ g(x) denotes limx→∞f(x)/g(x) = 1, f(x) ≍ g(x) means 0 < lim infx→∞f(x)/g(x)≤lim supx→∞f(x)/g(x)<∞.
2
Main results
Instead of focusing only on the sum of record values, we shall consider asymptotic distributions and Berry-Esseen bounds for a general randomly weighted sum.
Let {Un, n≥1} be independent uniformly distributed random variables, and{Yn, n≥1} be i.i.d.
non-negative random variables withEYi= 1 and Var(Yi) =c20. DenoteSn=Pni=1Yiand assume that
(U1,· · ·, Un) andSn+1 are independent for every fixedn. Letψ: [0,∞)→[0,∞) be a non-decreasing
continuous function. Put
Wn= n
X
i=1
ψ(UiSn+1)
and define
α(x) = Z 1
0 ψ(ux)du, β2(x) =
Z 1
0 (ψ(ux)−α(x))
2du=Z 1 0 ψ
2(ux)du
−α2(x), γ(x) =
Z 1
0
ψ3(ux)du.
Let
σ2n=β2(n+ 1) +n(n+ 1)c20[α′(n+ 1)]2 ,
whereα′(x) = dxdα(x) .
(A1) ∀ a >1, lim
n→∞|x−nsup|≤a√ n
γ(x)
n1/2β3(x) = 0,
(A2) ∀ a >1, lim
n→∞ 1
nσ2 n
sup |x−n|≤a√n
ψ4(x) β2(x) = 0,
(A3) ∀ a >1, lim
n→∞
ψ(n) +Ra√n
−a√n|ψ(x+n)−ψ(n)|dx
√
nσn
= 0. Then we have
Wn−nα(n+ 1)
√nσ
n
d.
−→N(0,1) (2.1)
Next theorem provides a Berry-Esseen bound.
Theorem 2.2 Assume that E(Y3
i )<∞ and that ψ is differentiable. Then
sup
x |P
³Wn−nα(n+ 1) √
nσn ≤
x´−Φ(x)| (2.2)
≤Cn−1/2³c0−3EY13+Rn+1,1+Rn+1,2+c0−2EY13Rn+1,3
´
for n≥(2c0)3, where C is an absolute constant,
Rn,1 = sup
|x−n|≤c0n2/3
γ(x)
β3(x), Rn,2 =
1
nα′(n) sup |x−n|≤c0n2/3
ψ2(x) β(x) ,
Rn,3 =
1
nα′(n) sup |x−n|≤c0n2/3
(ψ(x) +xψ′(x)).
The results given in Theorems 2.1 and 2.2 are especially appealing when ψ is a regularly varying function of orderθ. A Borel measurable function l(x) on (0,∞) is said to be slowly varying (at ∞) if
∀t >0, lim
x→∞
l(tx)
l(x) = 1.
ψ(x) is a regularly varying function of order θ if ψ(x) = xθl(x), where l(x) is slowly varying. It is known that forθ >−1 (see, for example, [2])
lim
x→∞ R∞
0 ψ(u)du xψ(x) =
1 1 +θ.
(2.3)
In particular, as x→ ∞ forθ >0
α(x) ∼ ψ(x) 1 +θ,
α′(x) ∼ θ 1 +θ
ψ(x)
x ,
(2.5)
β(x) ∼ θ
(1 +θ)√1 + 2θψ(x),
(2.6)
γ(x) ≤ 1 1 + 3θψ
3(x),
(2.7)
σn2 ∼ θ 2
(1 +θ)2(
1 1 + 2θ +c
2
0)ψ2(n).
(2.8)
Thus, we have
Theorem 2.3 Let ψ(x) = xθl(x), where θ > 0 and l(x) is slowly varying. Then (2.1) holds. In
addition if E(Yi3)<∞, and |ψ′(x)| ≤c1xθ−1l(x) for x >1, then
sup
x |P
³Wn−nα(n+ 1) √
nσn ≤
x´−Φ(x)| ≤An−1/2,
(2.9)
where A is a constant depending only on c0, c1 and EY13.
Another special case is ψ(x) = exp(c xτl(x)).
Theorem 2.4 Assume thatψ(x) = exp(cxτl(x)), wherec >0,0< τ <1/2andl(x)is slowly varying
at ∞. Assume that there exist 0< c1 ≤c2 <∞ and x0 >1 such that
c1xτ−1l(x)≤(xτl(x))′ ≤c2xτ−1l(x) for x≥x0.
(2.10)
Then
sup
x |P
³Wn−nα(n+ 1) √
nσn ≤
x´−Φ(x)| ≤An−1/2+τl(n),
(2.11)
where A is a constant depending only on c0, c1, c2, x0, τ and EY13.
Remark 2.1 Hu, Su and Wang (2002) recently proved that if τ > 1/2, there is no standardized
Wn that has non-degenerate limiting distribution, which in turn indicates that assuming τ < 1/2 in
Theorem 2.4 is necessary.
3
An application to the sum of record values
As we have seen in Section 1, the sum of record values Tn is a special case of Wn with Yi i.i.d.
exponentially distributed random variables. So, Theorems 2.1, 2.2 and 2.3 hold with c0 = 1. When
ψ(x) =xorψ(x) = lnx, Arnold and Villasenor (1998) proved the asymptotic normality forTnand also
proved that the limiting distribution is not normal if ψ(x) = 1−exp(−x/β). They then conjectured that the central limit theorem holds ifψ(x)∼xθlnv(x),θ≥0, v≥0, θ+v >0. Theorem 2.3 already implies that their conjecture is true if θ >0. Next theorem confirms that it is also true ifθ= 0 and
Theorem 3.1 Let ψ(x) = lnv(1 +x), v >0. Then
From (3.4) and (3.5) it follows that
Thus, by the above estimates
Rn,1 ≤Aln3(1 +n), Rn,2≤Aln2(1 +n), Rn,3≤Aln(1 +n).
(3.7)
This proves Theorem 3.1, by Theorem 2.2 and (3.7).
4
Proofs
The proofs are based on the fact that givenSn+1,{ψ(UiSn+1),1≤i≤n}is a sequence of i.i.d. random
variables with mean α(Sn+1) and variance β(Sn+1), and that Sn+1/(n+ 1)→1 with probability one
by the law of large numbers. Let
Hn=
Lemma 4.1 We have
and
Proof of Theorem 2.1. By (4.2), it suffices to show that
Hn,1 −→d. N(0,1)
Let Z1 and Z2 be independent standard normal random variables independent of {Ui, i ≥ 1} and
asn→ ∞. It is easy to see that
As to (4.11), observe that conditioning onSn+1,Hnis a standardized sum of i.i.d. random variables
and hence
Lemma 4.2 [Chen and Shao (2003)] Let {ξi,1≤i≤n} be independent random variables, gi :R1 →
Proof of Theorem 2.2. In what follows, we use C to denote an absolute constant, but its value could be different from line to line. By the Rosenthal inequality, we have
To apply Lemma 4.2, let gi(Yi) = (Yi−1)/(c0√n+ 1), S(i) =Sn+1−Xi and
∆i =
( ¯β(S(i))−β¯(n+ 1))Z1 c0α′(n+ 1)pn(n+ 1)
+ 1
c0α′(n+ 1)√n+ 1
Z S(i)
n+1(α
′(t)−α′(n+ 1))dt
for 1≤i≤n+ 1. Following the proof of (4.21), we have
E(|gi(Yi)(∆−∆i)| |Z1)
(4.22)
≤ |Z1|E|gi(Yi)( ¯β(Sn+1)−β¯(S
(i)))| c0nα′(n+ 1)
+ 1
c0α′(n+ 1)√n
Engi(Yi)
Z S¯n+1
¯ S(i) |α
′(t)−α′(n+ 1)|dto ≤ C|Z1|n−3/2Rn+1,2+CEY13c−02n−2Rn+1,3.
Letting
Hn,3(Z1, Z2) =
c0α′(n+ 1)pn(n+ 1) σn
³
Z2+
¯
β(n+ 1)Z1 c0α′(n+ 1)p
n(n+ 1) ´
and applying Lemma 4.2 for givenZ1 yields sup
x |P(Hn,3(Z1)≤x)−P(Hn,3(Z1, Z2)≤x)|
(4.23)
≤ Cn−1/2c−03EY13+Cn−1/2Rn+1,2E|Z1|+CEY13c0−2n−1/2Rn+1,3
≤ Cn−1/2³c−03EY13+Rn+1,2+c−02EY13Rn+1,3
´
.
On the other hand, it is easy to see that Hn,3(Z1, Z2) has the standard normal distribution. This
proves (4.19), as desired.
Proof of Theorem 2.4. Letρ(x) =c xτl(x). It is easy to see that condition (2.10) implies for
a >0
Z x
0 e aρ(t)dt
≍ ρ(xx)eaρ(x)
(4.24)
asx→ ∞, whereg(x)≍h(x) denotes 0<lim infx→∞g(x)/h(x)≤lim supx→∞g(x)/h(x)<∞. Thus, we have
α(x) ≍ 1
ρ(x)e
ρ(x), α′(x)≍ 1
xe ρ(x),
(4.25)
β2(x) ≍ 1
ρ(x)e
2ρ(x), γ(x)≍ 1 ρ(x)e
3ρ(x)
Letdn=n/ρ(n). Then
Following the proof of Theorem 2.2, we have sup
x |P
³Wn−nα(n+ 1) √
nσn ≤
x´−Φ(x)| (4.26)
≤ A(nτ−1/2l(n))2+An−1/2³R∗n,1+Rn,∗2+R∗n,3´,
where
R∗n,1 = sup |x−n−1|≤dn
γ(x)
β3(x), R∗n,2 =
1
nα′(n) sup |x−n−1|≤dn
ψ2(x) β(x) ,
R∗n,3 = 1
nα′(n) sup |x−n−1|≤dn
(ψ(x) +xψ′(x)).
Note that forx satisfying |x−n−1| ≤dn, by (2.10)
|ρ(x)−ρ(n)| ≤A1|x−n|ρ(n)/n≤A2dnρ(n)/n≤A3,
whereA1, A2, A3 denote constants that do not depend on n. Hence ψ(x) =ψ(n)eρ(x)−ρ(n)≍ψ(n),
which combines with (4.25) gives
R∗n,1 ≤Aρ1/2(n), R∗n,2 ≤Aρ1/2(n), R∗n,3 ≤Aρ(n).
(4.27)
5
Appendix
Proof of Lemma 4.1. Rewrite
α(x) = 1
x
Z x
0 ψ(u)du.
Recall that ψ(x) is non-decreasing and continuous. We have
α′(x) = −1
This proves Lemma 4.1.
References
[1] Arnold, B.C. and Villasenor, J.A.(1998). The asymptotic distributions of sums of records,
[2] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation. Cambridge University Press, Cambridge.
[3] Chen, L.H.Y. and Shao, Q.M.(2003). Uniform and non-uniform bounds in normal approxi-mation for nonlinear Statistics. Preprint.
[4] de Haan, L and Resnick, S.I. (1973b). Almost sure limit points of record values. J. Appl. Probab.10, 528–542.
[5] Embrechts, P., Kl¨uppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
[6] Hu, Z.S., Su, C. and Wang, D.C. (2002). Asymptotic distributions of sums of record values for distributions with lognormal-type tails.Sci. China Ser. A45, 1557–1566.
Preprint.
[7] Mikosch, T. and Nagaev, A.V. (1998). Large Deviations of heavy-tailed sums with applica-tions in insurance.Extremes 1, 81-110.
[8] Petrov, V.V.(1995).Limit Theorems of Probability Theory, Sequences of Independent Random Variables. Clarendon Press, Oxford.
[9] Resnick, S.I.(1973). Limit laws for record values. Stoch. Process. Appl.1, 67-82.
[10] Su, C. and Hu, Z.S.(2002). The asymptotic distributions of sums of record values for distribu-tions with regularly varying tails.J. Math. Sci. (New York)111, 3888–3894.
[11] Tata, M.N.(1969). On outstanding values in a sequence of random variables.Z. Wahrsch. Verw. Gebiete12, 1969 9–20.
Qi-Man Shao
Department of Mathematics University of Oregon
Eugene, OR 97403 USA
Department of Mathematics
Department of Statistics & Applied Probability National University of Singapore
Department of Statistics and Finance
University of Science an Technology of China Hefei, Anhui 230026
P.R. China
Gang Wei
Department of Mathematics Hong Kong Baptist University Kowloon Tong
